Circles and Segments related to Circles

Contributed by:

Sharp Tutor Mon, Jan 17, 2022 08:19 PM UTC

Some questions to be answered:
1. How do I identify segments and lines related to circles?
2. How do I use the properties of a tangent to a circle?

1. Circles
Chapter 10

2. Essential Questions
How do I identify segments and lines
related to circles?
How do I use properties of a tangent to a
circle?

3.  A circle is the set of all points in a plane that are
equidistant from a given point called the center of the
circle.
 Radius – the distance from the center to a point on the
circle
 Congruent circles – circles that have the same radius.
 Diameter – the distance across the circle through its
center

4. Diagram of Important Terms
center
radius
P
diameter
name of circle: P

5.  Chord – a segment whose endpoints are points on the
circle.
B
A
AB is a chord

6.  Secant – a line that intersects a circle in two points.
N
M
MN is a secant

7.  Tangent – a line in the plane of a circle that intersects
the circle in exactly one point.
T
S
ST is a tangent

8. Example 1
 Tell whether the line or segment is best described as a chord, a
secant, a tangent, a diameter, or a radius.
H
a. AH tangent
b. EI diameter
B E
C F
c. DF chord
I G
d. CE radius
A D

9. Tangent circles – coplanar circles that
intersect in one point

10. Concentric circles – coplanar circles that
have the same center.

11. Common tangent – a line or segment that is
tangent to two coplanar circles
 Common internal tangent – intersects the segment
that joins the centers of the two circles
 Common external tangent – does not intersect the
segment that joins the centers of the two circles
common external tangent
common internal tangent

12. Example 2
 Tell whether the common tangents are internal or external.
a. b.
common internal tangents common external tangents

13. More definitions
Interior of a circle – consists of the points
that are inside the circle
Exterior of a circle – consists of the points
that are outside the circle

14.  Point of tangency – the point at which a tangent line
intersects the circle to which it is tangent
point of tangency

15. Perpendicular Tangent Theorem
 If a line is tangent to a circle, then it is perpendicular to
the radius drawn to the point of tangency.
l
P
Q
If l is tangent to Q at P, then l  QP.

16. Perpendicular Tangent Converse
 In a plane, if a line is perpendicular to a radius of a circle
at its endpoint on the circle, then the line is tangent to
the circle.
l
P
Q
If l  QP at P, then l is tangent to Q.

17. Right Triangles
Pythagorean Theorem
Radius is perpendicular to the
tangent.  < E is a right angle
C
43
E
45
11
D

18. Example 3
C
Tell whether CE is tangent to D. 43
E
Use the converse of the Pythagorean 45
Theorem to see if the triangle is right. 11
D
11 + 43 ? 45
2 2 2
121 + 1849 ? 2025
1970  2025
CED is not right, so CE is not tangent to D.

19. Congruent Tangent Segments Theorem
 If two segments from the same exterior point are tangent
to a circle, then they are congruent.
R
P
S
T
If SR and ST are tangent to P, then SR  ST.

20. Example 4
AB is tangent to C at B. D
AD is tangent to C at D. x2 + 2
Find the value of x. C A
11
AD = AB
B
x2 + 2 = 11
x2 = 9
x = 3

21.  Central angle – an angle whose vertex is the center of a circle.
central angle

22. Minor arc – Part of a circle that measures less
than 180°
Major arc – Part of a circle that measures
between 180° and 360°.
Semicircle – An arc whose endpoints are the
endpoints of a diameter of the circle.
Note : major arcs and semicircles are named with
three points and minor arcs are named
with two points

23. Diagram of Arcs
A
minor arc: AB
major arc: ABD
D B
C
semicircle: BAD

24. Measure of a minor arc – the measure of its
central angle
Measure of a major arc – the difference between
360° and the measure of its associated minor
arc.

25. Arc Addition Postulate
 The measure of an arc formed by two adjacent arcs is the sum of
the measures of the two arcs.
A
C
mABC = mAB + mBC
B

26.  Congruent arcs – two arcs of the same circle or of
congruent circles that have the same measure

27. Arcs and Chords Theorem
 In the same circle, or in congruent circles, two minor arcs are
congruent if and only if their corresponding chords are
congruent.
A
B
AB  BC if and only if AB  BC
C

28. Perpendicular Diameter Theorem
 If a diameter of a circle is perpendicular to a chord, then the
diameter bisects the chord and its arc.
F
DE  EF, DG  FG
E
G
D

29. Perpendicular Diameter Converse
 If one chord is a perpendicular bisector of another chord, then
the first chord is a diameter.
J
M
K
L
JK is a diameter of the circle.

30. Congruent Chords Theorem
 In the same circle, or in congruent circles, two chords are
congruent if and only if they are equidistant from the center.
C
G
AB  CD if and only if EF EG. E D
B
F
A

31. Example 1
 Find the measure of each arc.
a. LM 70°
N L
P 70
b. MNL 360° - 70° = 290°
c. LMN 180° M

32. Example 2
 Find the measures of the red arcs. Are the arcs congruent?
A
C
41
41
D
mAC = mDE = 41 E
Since the arcs are in the same circle, they are congruent!

33. Example 3
 Find the measures of the red arcs. Are the arcs congruent?
A
D
81
E
C
mDE = mAC = 81
However, since the arcs are not of the same circle or
congruent circles, they are NOT congruent!

34. Example 4
B
Find mBC.
(3x + 11) 
(2x + 48)
3x + 11 = 2x + 48
A
x = 37
D C
mBC = 2(37) + 48
mBC = 122

35.  Inscribed angle – an angle whose vertex is on a circle
and whose sides contain chords of the circle
 Intercepted arc – the arc that lies in the interior of an
inscribed angle and has endpoints on the angle
intercepted arc
inscribed angle

36. Measure of an Inscribed Angle Theorem
 If an angle is inscribed in a circle, then its measure is
half the measure of its intercepted arc.
A
1 C
mADB = mAB D
2 B

37. Example 1
 Find the measure of the blue arc or angle.
E
a. S R b.
80
F
Q G
T
1
mQTS = 2(90 ) = 180 mEFG = (80 ) = 40
2

38. Congruent Inscribed Angles Theorem
If two inscribed angles of a circle intercept
the same arc, then the angles are
congruent.
A
B
C
D
C   D

39. Example 2
It is given that mE = 75 . What is mF?
Since E and F both intercept D
the same arc, we know that the
angles must be congruent.
E
mF = 75
F
H

40.  Inscribed polygon – a polygon whose vertices all lie on a
circle.
 Circumscribed circle – A circle with an inscribed polygon.
The polygon is an inscribed polygon and
the circle is a circumscribed circle.

41. Inscribed Right Triangle Theorem
 If a right triangle is inscribed in a circle, then the hypotenuse
is a diameter of the circle. Conversely, if one side of an
inscribed triangle is a diameter of the circle, then the triangle
is a right triangle and the angle opposite the diameter is the
right angle.
A
B is a right angle if and only if AC
is a diameter of the circle. B
C

42. Inscribed Quadrilateral Theorem
 A quadrilateral can be inscribed in a circle if and only if
its opposite angles are supplementary.
E
F
C
D
G
D, E, F, and G lie on some circle, C if and only if
mD + mF = 180 and mE + mG = 180 .

43. Example 3
 Find the value of each variable.
D
a. b.
B z
G y 120 E
Q
A 2x
80
F
C
mD + mF = 180 mG + mE = 180
2x = 90
z + 80 = 180 y + 120 = 180
x = 45
z = 100 y = 60

44. Tangent-Chord Theorem
 If a tangent and a chord intersect at a point on a circle,
then the measure of each angle formed is one half the
measure of its intercepted arc.
B
1
m1 = mAB C
2
1 1
m2 = mBCA 2
2 A

45. Example 1
Line m is tangent to the circle. Find mRST m
R
102
mRST = 2(102 )
S
mRST = 204
T

46. Try This!
Line m is tangent to the circle. Find m1
1 R
m1 = (150 )
2 1
m
m1 = 75
150
T

47. Example 2
BC is tangent to the circle. Find mCBD. C
A
(9x+20)
5x B
2(5x) = 9x + 20
10x = 9x + 20
x = 20
D
mCBD = 5(20 )
mCBD = 100

48. Interior Intersection Theorem
 If two chords intersect in the interior of a circle, then the
measure of each angle is one half the sum of the
measures of the arcs intercepted by the angle and its
vertical angle.
1 D
m1 = (mCD + mAB) A
2
1
2
1
m2 = (mAD + mBC) C
2 B

49. Exterior Intersection Theorem
If a tangent and a secant, two tangents, or
two secants intersect in the exterior of a
circle, then the measure of the angle
formed is one half the difference of the
measures of the intercepted arcs.

50. Diagrams for Exterior
Intersection Theorem
B
A P
1
2
Q
C R
1
1 m2 = (mPQR - mPR)
m1 = (mBC - mAC) X 2
2
W
3
Z Y
1
m3 = (mXY - mWZ)
2

51. Example 3
P
 Find the value of x. 106
Q
1
x = (mPS + mRQ)
2 x
1 S
x = (106 +174 )
2 174 R
1
x= (280)
2
x = 140

52. Try This!
 Find the value of x. T
40
1 S
x = (mST + mRU) x
2
U
1
x = (40 +120 )
2
R 120
1
x= (160)
2
x = 80

53. Example 4
 Find the value of x.
1
72 = (200 - x ) 200
2
144 = 200 - x
x 72
x = 56

54. Example 5
 Find the value of x.
A
mABC = 360 - 92 B
mABC = 268 92 x
1
x= (268 - 92) C
2
1
x = (176)
2
x = 88

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